Modified regularization method for a Cauchy problem of the Laplace equation

Abstract

In this paper, we consider the Cauchy problem for the Laplace's equation in a rectangle, where the Cauchy data is given for y = 0 and boundary data are given for x = 0 and x = π. The solution is sought in the interval 0 < y ≤ 1. A modified method is presented and error estimate is obtained. Combining the method of modified, we implement the numerical solution. A numerical example shows that the method works well.